3 Short Term Trend Analysis
Description: Time series trend analysis for short time series
Found in: State of the Ecosystem - Gulf of Maine & Georges Bank (2025+), State of the Ecosystem - Mid-Atlantic (2025+)
Indicator category: Extensive analysis, not yet published
Contributor(s): Andy Beet
Data steward: NA
Point of contact: Andy Beet, andrew.beet@noaa.gov
Public availability statement: NA
3.1 Methods
In prep: A.Beet “A test for short term trend detection in the presence of autocorrelation”
The specific model addressed here is of the form,
\[\begin{equation} Y_t = \beta_0 + \beta_1 t + \epsilon_t \end{equation}\]
where \(\epsilon_t = \phi\epsilon_{t-1} + z_t\) is a stationary first order autoregressive process with \(z_t \sim N(0,\sigma^2)\). Interest centers on testing the null hypothesis, \(H_0:\beta_1 = 0\) against the alternative, \(H_1:\beta_1 \neq 0\)
Testing for a trend in time series data has been addressed by many authors from a wide range of disciplines including economics, statistics, hydrology, ecology, fisheries, and epidemiology (Cochrane and Orcutt (1949), Prais and Winsten (1954), Beach and MacKinnon (1978), Park and Mitchell (1980), Brillinger (1994), Bence (1995), Woodward, Bottone, and Gray (1997), X. Zhang et al. (2000), Yue et al. (2002), Wang et al. (2015) ,Hardison et al. (2019b)). These approaches have typically taken one of three paths; non parametric methods such as the Mann Kendall test and its pre-whitening variants (Hamed and Ramachandra Rao (1998), X. Zhang et al. (2000), Yue and Wang (2002), Wang et al. (2015)); parametric methods involving data transformation such as Cochrane and Orcutt (1949), Prais and Winsten (1954), Woodward, Bottone, and Gray (1997); parametric methods such as generalized least squares and maximum likelihood estimation (Beach and MacKinnon (1978), Davidson and Mackinnon (1999), J. C. Pinheiro and Bates (2000)).
It has been well documented that under the null hypothesis of no trend, \(H_0:\beta_1=0\), in the presence of autocorrelation, parametric tests relying on asymptotic distribution theory reject the null hypothesis too frequently, leading to nominal significance levels that are too high, even for relatively long time series of length n = 100 (Woodward, Bottone, and Gray (1997)). Non parametric tests like those listed above also suffer the same problem.
We introduce a test that, like Beach and MacKinnon (1978), uses maximum likelihood for parameter estimation, but differs in that the significance of the likelihood ratio statistic, LR, is assessed via a parametric bootstrap (Efron and J. (1993)). Parametric bootstrap procedures have been used in some of the aforementioned work. Woodward, Bottone, and Gray (1997) uses an alternative statistic, the Cochrane-Orchutt statistic, for detecting a trend. Rayner (1990) focuses on the significance of the AR(1) parameter and Bence (1995) focuses on adjusting confidence intervals. We use the parametric bootstrap as an alternative means of assessing the significance of the LR statistic.
The Likelihood ratio statistic combined with a parametric bootstrap is employed to test for a linear trend in the presence of autocorrelation in the form of an AR(1) process. Small samples of size, n = 10, are of particular interest.
3.1.3 Data analysis
Code used for the fitting and evaluation of short term trend can be found here.
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